#6903

order	243
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	Aut(G) = C₉×AGL(2,𝔽₃) (order 3888) · Inn(G) = C₃² · Out(G) = C₉×GL(2,𝔽₃)
homology	Abelianization (H₁) = C₃²×C₉ · Schur multiplier (H₂) = C₃²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂₇	C₃²
commutator, socle	C₃	C₃²×C₉
Frattini, cosocle	C₉	C₃³
prime radical	H(3,𝔽₃)	C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂₇ ◁ G
LECS	G ▷ C₉ ▷ C₃ ▷ C₁

55 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 81

A minimal faithful permutation representation — 3 generators on 81 points:

(1,2,6,4,8,18,14,24,39,5,9,19,15,25,40,33,48,62,16,26,41,34,49,63,55,69,77)(3,7,17,12,22,37,31,46,60,13,23,38,32,47,61,54,68,76,11,21,36,30,45,59,53,67,75)(10,20,35,28,43,57,51,65,73,27,42,56,50,64,72,70,78,80,29,44,58,52,66,74,71,79,81);(1,3,10)(2,7,20)(4,12,28)(5,13,27)(6,17,35)(8,22,43)(9,23,42)(11,29,16)(14,31,51)(15,32,50)(18,37,57)(19,38,56)(21,44,26)(24,46,65)(25,47,64)(30,52,34)(33,54,70)(36,58,41)(39,60,73)(40,61,72)(45,66,49)(48,68,78)(53,71,55)(59,74,63)(62,76,80)(67,79,69)(75,81,77);(3,11,13)(7,21,23)(10,27,29)(12,30,32)(17,36,38)(20,42,44)(22,45,47)(28,50,52)(31,53,54)(35,56,58)(37,59,61)(43,64,66)(46,67,68)(51,70,71)(57,72,74)(60,75,76)(65,78,79)(73,80,81)

Polycyclic presentation:

3^5|a^3=d,d^3=e,c^b=ce

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#92318⋊C₂, #92318⋊C₄, #14517142, …
Fitting subgroup of	#92318, #92319, #11757686, …
maximal subgroup of	#92318, #92319, C₂×#6903, …
maximal normal subgroup of	#92318, #92319, C₂×#6903, …
maximal quotient of	C₂×#6903, #10624681, #10624684, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(243, 50);